Degenerate form

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[edit] Definition

In mathematics, specifically linear algebra, a degenerate bilinear form f(x,y) on a vector space V is one such that the map from V to V * (the dual space of V) given by v \mapsto f(-,v) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that

f(x,y)=0\, for all y \in V.

A nondegenerate form is one that is not degenerate, meaning that v \mapsto f(-,v) is an isomorphism, or equivalently in finite dimensions, if and only if

f(x,y)=0\, for all y \in V implies that x = 0.

There is the closely related notion of a perfect pairing; these agree over fields but not over general rings.

The most important examples of nondegenerate forms are inner products and symplectic forms.

[edit] Infinite dimensions

Note that in an infinite dimensional space, we can have a bilinear form f for which v \mapsto f(-,v) is injective but not surjective (confer Hilbert's paradox). For example, on the space of continuous functions on a closed bounded interval, the form

f(\phi,\psi) = \int\psi(x)\phi(x) dx

is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form. On the other hand, this bilinear form satisfies

f(\phi,\psi)=0\, for all \phi\, implies that \psi=0.\,

[edit] Terminology

If f vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form f on V the set of vectors

\{x\in V \mid f(x,y) = 0 \mbox{ for all } y \in V\}

forms a totally degenerate subspace of V. f is nondegenerate if and only if this subspace is trivial.

If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the associated matrix is non-singular. These statements are independent of the chosen basis.

Sometimes the words anisotropic, isotropic and totally isotropic are used for nondegenerate, degenerate and totally degenerate respectively, although definitions of these latter words can vary slightly between authors. Nondegenerate bilinear forms are also sometimes called perfect pairings.

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